Results 1  10
of
37
Heat flow on Finsler manifolds
, 2009
"... This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : TxM → R+ on each tangent space. Mostly, we will require that this norm is strongly convex and smooth and that it depends smoothly on the base point x. The particu ..."
Abstract

Cited by 47 (18 self)
 Add to MetaCart
(Show Context)
This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : TxM → R+ on each tangent space. Mostly, we will require that this norm is strongly convex and smooth and that it depends smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: • either as gradient flow on L2 (M,m) for the energy E(u) = 1
Heat flow on Alexandrov spaces
, 2012
"... We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L²space produces the same evolution as the gradient flow of the relative entropy in the L²Wasserstein space. This means that the heat flow is well defined by either one of the t ..."
Abstract

Cited by 42 (15 self)
 Add to MetaCart
We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L²space produces the same evolution as the gradient flow of the relative entropy in the L²Wasserstein space. This means that the heat flow is well defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as BakryÉmery gradient estimates and the Γ2condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift.
Some geometric calculations on Wasserstein space
, 2007
"... We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold. ..."
Abstract

Cited by 23 (2 self)
 Add to MetaCart
We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.
The heat equation on manifolds as a gradient flow in the Wasserstein space.
 Ann. Inst. Henri Poincare Probab. Stat.,
, 2010
"... Abstract. We study the gradient flow for the relative entropy functional on probability measures over a Riemannian manifold. To this aim we present a notion of a Riemannian structure on the Wasserstein space. If the Ricci curvature is bounded below we establish existence and contractivity of the gr ..."
Abstract

Cited by 19 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We study the gradient flow for the relative entropy functional on probability measures over a Riemannian manifold. To this aim we present a notion of a Riemannian structure on the Wasserstein space. If the Ricci curvature is bounded below we establish existence and contractivity of the gradient flow using a discrete approximation scheme. Furthermore we show that its trajectories coincide with solutions to the heat equation. Résumé. Nous étudions les flux gradients dans l'espace des mesures de probabilité sur une variété Riemannienne pas nécessairement compacte. Dans ce but nous munissons l'espace de Wasserstein avec une sorte de structure Riemannienne. Si la courbure de Ricci de la variété est bornée inférieurement nous démontrons qu'il existe un flux gradient contractif pour l'entropie relative. Il est construit explicitement en utilisant une approximation variationelle discrète. De plus ses trajectoires Coïncident avec les solutions à l'équation de la chaleur. MSC: 35A15; 58J35; 60J60
On the inverse implication of BrenierMcCann theorems and the structure of (P2(M), W2
"... We do three things. First, we characterize the class of measures µ ∈ P2(M) such that for any other ν ∈ P2(M) there exists a unique optimal transport plan, and this plan is induced by a map. Second, we study the tangent space at any measure and we identify the class of measures for which the tangent ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
(Show Context)
We do three things. First, we characterize the class of measures µ ∈ P2(M) such that for any other ν ∈ P2(M) there exists a unique optimal transport plan, and this plan is induced by a map. Second, we study the tangent space at any measure and we identify the class of measures for which the tangent space is an Hilbert space. Third, we prove that these two classes of measures coincide. This answers a question recently raised by Villani. Our results concerning the tangent space can be extended to the case of Alexandrov spaces.
Finsler interpolation inequalities
, 2009
"... We extend CorderoErausquin, McCann and Schmuckenschläger’s Riemannian BorellBrascampLieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvaturedimension condition and a certain lower Ricci curvature bound. We also prove a ne ..."
Abstract

Cited by 17 (3 self)
 Add to MetaCart
We extend CorderoErausquin, McCann and Schmuckenschläger’s Riemannian BorellBrascampLieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvaturedimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest.
Total Curvatures of Model Surfaces Control Topology of Complete Open Manifolds with Radial Curvature Bounded Below. I
, 2009
"... We investigate the finiteness structure of a complete open Riemannian nmanifold M whose radial curvature at a base point of M is bounded from below by that of a noncompact von Mangoldt surface of revolution with its total curvature greater than π. We show, as our main theorem, that all Busemann f ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
(Show Context)
We investigate the finiteness structure of a complete open Riemannian nmanifold M whose radial curvature at a base point of M is bounded from below by that of a noncompact von Mangoldt surface of revolution with its total curvature greater than π. We show, as our main theorem, that all Busemann functions on M are exhaustions, and that there exists a compact subset of M such that the compact set contains all critical points for any Busemann function on M. As corollaries by the main theorem, M has finite topological type, and the isometry group of M is compact.
First variation formula in Wasserstein spaces over compact Alexandrov spaces
, 2010
"... We extend results proven by the second author ([Oh]) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces X with curvature bounded below: the gradient flow of a geodesically convex functional on the quadratic Wasserstein space (P(X), W2) satisfies the evolution variational ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
We extend results proven by the second author ([Oh]) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces X with curvature bounded below: the gradient flow of a geodesically convex functional on the quadratic Wasserstein space (P(X), W2) satisfies the evolution variational inequality. Moreover, the gradient flow enjoys uniqueness and contractivity. These results are obtained by proving a first variation formula for the Wasserstein distance. 1